3D shapes formula chart
Volume and surface area formulas for the common three-dimensional solids. Volume measures the space inside, in cubic units; surface area measures the outside skin, in square units.
Volume and surface area
| Solid | Volume | Surface area |
|---|---|---|
| Cube | s³ | 6s² |
| Rectangular prism | l × w × h | 2(lw + lh + wh) |
| Cylinder | πr²h | 2πr² + 2πrh |
| Sphere | (4/3)πr³ | 4πr² |
| Cone | (1/3)πr²h | πr² + πrℓ |
| Square pyramid | (1/3)b²h | b² + 2bℓ |
| Triangular prism | (½ × b × h) × length | 2(base area) + (perimeter × length) |
In the cone and pyramid formulas, the symbol ℓ is the slant height, the distance along the sloping face, which differs from the vertical height h. Volume is always in cubic units and surface area in square units. The one-third factor on the pointed solids comes from the geometry of cones and pyramids.
Need volumes or areas?
See the Volume Formula Chart and the Area Formula Chart.
Volume versus surface area
Volume tells you how much a solid holds; surface area tells you how much material covers it. A cylinder volume is its circular base area times its height, while its surface area is the two circular ends plus the rectangle that wraps around the side, which unrolls to 2 pi r times h. Keeping the two ideas separate avoids mixing square and cubic units.
Slant height versus vertical height
For cones and pyramids, two different heights appear. The vertical height h runs straight up from the base to the apex and is used for volume. The slant height, written here as the script letter l, runs along the sloping surface and is used for the lateral surface area. They are linked by the Pythagorean theorem.
FAQ
What is the volume of a sphere?
(4/3)πr³. For a radius of 3, that is about 113 cubic units.
What is the surface area of a cube?
6s², since a cube has six identical square faces of side s.
What is slant height?
The distance along the sloping face of a cone or pyramid, from the base edge to the apex, used in surface-area formulas.